J . Phys. Chem. 1992, 96, 3963-3969
3963
Electrostatic Persistence Length of a Smoothly Bending Polyion Computed by Numerical Counterion Condensation Theory‘ Marcia 0. Fenley, Gerald S.Manning,* and Wilma K. Olson Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick, New Jersey 08903 (Received: September 20, 1991)
Using numerical counterion condensation theory, we compute the electrostatic component of the persistence length of DNA for three models. In the line model, the phosphate charges are represented as points uniformly distributed on a line of zero radius. In the helical model, the phosphates are given the three-dimensional coordinates deduced from X-ray fiber diffraction; the helical model is considered with and without the dielectric saturation effect. For each model, the free energy of bending smoothly to a specified radius of curvature is calculated. All models exhibit strong end effects for finite-length DNA. Our most reliable model is the helical model with dielectric saturation. Persistence lengths calculated for it are linearly correlated, to good numerical approximation, with the inverse concentration of salt. Comparison with experimental data suggests that real thermal bending fluctuationsof DNA in solution may not be describable as the smooth length-invariant bending postulated by any of the models.
Introduction When a linear polyion bends, the distances between pairs of charged groups on the polymer change. On one hand, since ionic forces are strong, this effect is expected to contribute substantially to the local stiffness of the polyion chain. On the other hand, one might anticipate some cancellation of competing trends, as distances between pairs of charged groups on one side of the polyion decrease when the chain is bent but increase on the other side. Experimental measurements indicate a strong decrease of the persistence length of charged polymers with increasing ionic ~ t r e n g t h . ~ .A~ consistent interpretation is that the decreased distance between some pairs of polymer charges outweighs the increased distance between others. On balance, electrostatic repulsion increases on bending and acts to resist the bend. The resistance becomes smaller as the concentration of screening salt increases. There are two classes of pertinent theories. One envisages the polymer as wormlike; the curvature along the chain is a smooth function of arc length along the c o n t o ~ r . ~The ? ~ ionic strength dependence of the persistence length predicted by this type of theory can be used as a reliable predictive basis for data on synthetic polyelectrolytes of relatively simple s t r u c t ~ r e .These ~ theories are less successful in predicting the persistence length of double-helical, base-pair-stacked DNA, as we will indicate in some detail in this paper. To account for the DNA data, a different kind of theory has been constructed, based on a model of bending as the result of discontinuous kinks, with chain segments between kinks slightly contra~ted.~.~ Another possibility that has not yet been analyzed is smooth bending coupled to a decrease in contour length. Still another is that seemingly minor deficiencies in existing smooth bending theories could account for the disagreement with data on DNA; indeed, bending is a process involving only small free-energy changes. We wish to explore this last hypothesis in the present paper. Current theories of the electrostatic contribution to the persistence length of a length-invariant, smoothly bending cylinder do not account for the detailed spatial distribution of the charges on the DNA. In this paper, we are able to use the phosphate coordinates deduced from X-ray fiber diffraction models. Current (1) Work supported by the US.Public Health Service, Grants GM36284 (G.S.M.), GM20861 (W.K.O.), and GM34809 (W.K.O.). M.O.F. is the recipient of a John von Neumann Fellowship from Rutgers University. Calculations were performed at the Rutgers Center for Computational Chemistry. (2) Borochov, N.; Eisenberg, H.; Kam, 2. Biopolymers 1981,20,231-235. (3) Tricot, M. Macromolecules 1984, 17, 1698-1704. (4) Le Bret, M. J. Chem. Phys. 1982, 76, 6243-6255. (5) Fixman, M. J. Chem. Phys. 1982, 76, 6346-6353. (6) Manning, G. S . Biopolymers 1983, 22, 689-729. (7) Manning, G. S . In Structure and Dynamics ofBiopofymers; Nicolini, C., Ed.; Martinus Nijhoff Dordrecht, Holland, 1987; pp 169-187.
theories are able to model the different dielectric properties of the DNA and the water in which it is immersed as a jump discontinuity between different but uniform dielectric constants inside and outside the cylinder. We have found,8 with Jayaram et al.? that dielectric saturation (which produces an effective dielectric constant that decreases with the distance between the pair of charges whose interaction it screens) may be of importance, and we are able to incorporate this effect into our calculations of the electrostatic persistence length. Another issue is the influence of end effects. Some experiments are designed to extract a persistence length from short DNA fragments.I0 This persistence length is not necessarily that of long DNA. The problem is to know when a fragment is sufficiently long to be in the polymer limit. Only an initial, approximate, analysis of this question is currently available (which, however, as we shall see, turns out to give reasonably good results).” Here, we present a detailed computation.
Model and Method We consider the interaction of counterions with the charged phosphate groups on DNA of varying curvature and length. The DNA molecule is modeled in two ways, first, as a finite line of evenly spaced charges (charge spacing 1.69 A), then, as a three-dimensional phosphate charge distribution. We will call the first model the line model, and the second, the helical model. For the helical model, the phosphorus coordinates of canonical B-DNA (Le., straight DNA), taken as the unit charge centers for the phosphate groups, are from X-ray fiber diffraction data of Arnott (unpublished results). These coordinates are located on a double helix with a radius of 9 8, and mean rise of 3.38 A per base pair. For both DNA models, the coordinates of the phosphorus atoms for the smoothly bent DNA are obtained by a homogeneous coordinate transformation. Let the original phosphorus coordinates of straight DNA be ( x y , z ) . The DNA is bent by the following smooth coordinate transformation and the resulting ( x ’ J ’ , ~ ? coordinates are given by x ’ = ( ~ + x ) c o s e y’= y z’= (R+x)sinO (1) where R is the radius of curvature, 0 = z/R, and the doublehelical axis is originally along the z axis. The double-helical axis is taken into a circular arc in the x’z’plane with center at (O,O,O) and with radius R (the bending is confined to a single plane, which is consistent with our subsequent use of very small bending angles ( 8 ) Fenley, M. 0.;Manning, G. S.; Olson, W. K. Biopolymers 1990, 30, 1191-1203. (9) Jayaram, B.; Swaminathan, S.; Beveridge, D. L.; Sharp, K.; Honig, B. Macromolecules 1990, 23, 3156-3165. (IO) Porschke, D. Biophys. Chem. 1991, 40, 169-179. (11) Hagerman, P. J. Biopolymers 1983, 22, 811-814.
0022-365419212096-3963%03.00/0 0 1992 American Chemical Societv
Fenley et al.
3964 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 in the regime of linearly elastic response). Therefore, a uniform radius of curvature R is assumed throughout the contour length L of the DNA (this bending process introduces a small uniform bending angle between any two adjacent base pairs). The values of R in our studies are in the range 400-2000 A. The bending angle for the smallest radius of curvature considered (R = 400 A) is roughly 0.5' per base pair. However, for the line model, values of R as small as 45 A were considered for the 60-mer DNA segment, which corresponds to a bending angle of 4.3O per base pair). Since for the smallest R considered (400 A), the local structure of DNA is virtually straight (no disruption of the standard bond angles and bond lengths), it is not necessary to relax the DNA structure by energy minimization (this consideration is only important for the helical model). The counterions are treated as point charges. The solvent is a dielectric continuum. For the solvent, we consider two continuum models. In one, the dielectric constant t is taken to be that of water, 78.3. This dielectric model will be called the constant dielectric model. The other model accounts for dielectric saturation near the centers of phosphate charges by use of the Hingerty-Lavery distance-dependent dielectric function t(r):l29l3
where q, is the dielectric constant of the pure solvent (78.3, in our case), r is the distance between phosphate charges, and S is the slope of the sigmoidal segment of the function. In accord with the experimental observation, the counterion binding fraction is nearly a constant over most of the salt concentration range for S = 0.16.* Therefore, the value of S is fixed at 0.16 in our calculations. This model will be called the dielectric saturation model. As we will see, in order to obtain the electrostatic persistence length we must first compute the ionic bending free energy, that is, the ionic free energy difference between the bent and straight DNA. The ionic free energy for each conformation, bent and straight, is computed within the framework of the numerical counterion condensation theory.* For clarity we present here the ionic free energy of a DNA segment containing P phosphate groups:
In
(YC,
+ N-I(l
- NON)C~) -
+ (N-I - 0,) + PI where q is the unit electrical charge, NONis the fractional extent of neutralization of the phosphate groups by condensed N-valent counterions (ON, for short, is the "binding fraction"), c, is the salt concentration (molarity), cpis the phosphate concentration, Y and 'v are the number of counterion and co-ions, respectively, in the formula for the simple salt, and c is the dielectric constant of the solvent (which is water in our case). For the constant dielectric model e is 78.3 and for the dielectric saturation model t is given by eq 2. The distance rkj is between phosphorus coordinates rk and rj of the kth and jth phosphate charges, k is the Boltzmann constant, and Tis the temperature, taken as 298 K. The Debye screening parameter is designated by K and is given by K~
= (87)
X
10-3N,,(q2/eokT)I
(4)
where NAvis Avogadro's number, to is 78.3, and I is the ionic strength. For our particular case of salt concentration in excess over phosphate concentration, I is proportional to the square root (12) Hingerty, B. E.; Ritchie, R. H.; Ferrell, T. L.; Turner, J. E. Biopolymers 1985, 24, 421-439.
(13) Lavery, R. In Unusual DNA Structures; Wells, R. D.,Harvey, S. C., Eds.; Springer-Verlag: New York, 1988; pp 189-206.
TABLE I: Effect of Varying Radius of Curvature R on the Sodium Binding Fraction 8 , of 600-bp DNA at Various Salt Concentrations C .a ~
~~
~
61
M 0.0001 c.,
0.0010
0.0100 0.1000
R = m (straight B-DNA) 0.7570 0.7614 0.7645 0.773 1
R = 2000 A (bent B-DNA) 0.7571 0.7614 0.7645 0.7731
R = 400 A (bent B-DNA) 0.7602 0.7618 0.7645 0.7731
'The helical axis of the straight DNA is smoothly bent into a circular arc of radius R. The solvent is considered as a uniform dielectric continuum (DC model). of salt concentration. In eq 3, u is the condensation volume per mole of phosphate groups. It is set equal to the line model value for B-DNA ( u = 635 cm3/mol phosphate groups). In the computation of we set cp = 0. We assume the delocalized mode of counterion binding. For further details and a more complete discussion of the underlying assumptions, the reader is referred elsewhere.* To calculate QhC, we first need to obtain the counterion binding fraction ON. This is accomplished by finding the root of the equation (~Qo~c/&9N)cp=o = 0. The left-hand side of this equation is given by
oon'c
where K is given by eq 4 and all other parameters have already been discussed. From the values of ON and u for the straight and bent DNA we can compute the ionic bending free energy AGionic= Gionic(bentDNA) - Gionic(straightDNA)
(6)
The ionic free energy required to bend a DNA segment of contour length L smoothly and isotropically to a finite radius of curvature R depends on the electrostatic persistence length P,: AGionic = P,k TL / 2R2 (7) This relation assumes a Hookean restoring force and uniform radius of curvature throughout the contour length. Also, it is strictly valid only in the limit of very large radius of curvature, since the right-hand side is actually an expansion in powers of 1/R2, with only the lowest order term retained. By considering finite values of the radius of curvature, we can examine the range of validity of eq I. To obtain P,, we first need to know the range of R where there is a linear dependence of AG with 1/R2. In this range of R we extract P, from the slope of a plot of A P versus ( 1/R2). We compute P, as a function of ionic strength and DNA segment length. (a) Line Model. In this section we treat DNA segments of varying length (60-600 base pairs) by means of the finite line charge model. The constant dielectric model (dielectric constant fixed at 78.3) is used for the solvent, since this dielectric, together with the line of charges, is known to give accurate results for the ionic properties of DNA. We begin by calculating univalent counterion binding fractions O1 for both straight and bent DNA. By computing the counterion binding fraction as a function of the radius of curvature we can examine the influence of the bending process on the extent of counterion binding. As shown in Table I the extent of Na+ binding to a 600-mer varies very slightly with decreasing values of the radius of curvature M NaCl, the Na+ binding fraction increases with R. At decreasing radius of curvature, which causes distances between pairs of phosphates to decrease. However, the increase of 8, at the smallest radius of curvature considered, R = 400 A, is minimal,
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 3965
Electrostatic Persistence Length
400
0 0.0001 M NaCl
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."..............
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'
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i
......................
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l/c, (1/M)
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Figure 1. Ionic bending free energy (AC) for a 600-base-pair DNA as a function of the square of the inverse of the radius of curvature ( R ) at various salt concentrations. The DNA is modeled as a finite line of charges. The constant dielectric model is employed for the solvent.
less than 1%. An even smaller increase of the Na' binding fraction is predicted for higher salt concentrations, which more efficiently screen the phosphate-phosphate repulsions. We have also checked the situation at a much smaller radius of curvature, comparable to that of nucleosomal DNA; the counterion binding fraction of a 60-mer (c, = M) with R = 45 A is larger relative to R = 2000 A by only 2%. Therefore, our results indicate that the Na' binding fraction is practically insensitive to changes in the radius of curvature. This trend is in accord with a theoretical study of odijk and MandelI4 which shows that the effective polyion charge (which includes the counterion condensation phenomenon) is not affected by the flexibility (bending) of the polyion. Our study is also in qualitative agreement with a recent Poisson-Boltzmann (PB) studyI5 of a polyelectrolyte modeled as a string of charged beads immersed in salt solution which predicts that the effective surface charge [not the same as our effective point charges q( 1 - NO,)] changes very little with decreasing radius of curvature (-2% for the value of the effective surface charge a t R = 100 A relative to R = m). Therefore, our results as well as those of others indicate that the curvature of the polyion can be neglected in the calculation of the counterion binding fraction (no significant extra accumulation of counterions occurs due to the smooth bending process). It is well known that Hooke's law given by eq 7 is strictly valid only in the limit of large radius of curvature. Figure 1 shows that the bending free energy AG for a 600-mer DNA is not generally a linear function of l / R Z the ; deviation from linearity is especially pronounced at low salt. For practical reasons we wanted to check the extent of violation of Hooke's law a t very small radius of curvature characteristic of nucleosomal DNA. Deviation from linearity is on the order of 25% for a 60-mer bent to R = 45 A (data not shown), not enough to change conclusions based on Hooke's law estimates of the energy stored when DNA is wrapped around the nucleosome core. The departure from Hookean behavior of the ionic bending free energy for small radius of curvature is also predicted by Fixman's PB study.I4 Since for the 600-mer AG varies linearly with 1/R2 for R > 700 A at all salt concentrations (within reasonable accuracy), we obtain the electrostatic persistence length P, by calculating the slope of the best line that passes through points ( 1/R2,AG) with (14) Odijk, T. J.; Mandel, M. Physica 1978, 9 3 4 298-306. (15) Fixman, M. J . Chem. Phys. 1990, 92, 6283-6293.
F i p e 2. Electrostatic persistence length P, for a 600-mer DNA as a function of the inverse of salt concentration c,. The dashed line is a least-squares fit of the results to a function of type y = ax b, where y and x represent P. and l/cs, respectively. The correlation coefficient is also reported.
+
3000
A 0.0001MNaCl
.
0 0.0005MNaCl
A 0.001 MNaCl
0
1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 1 0
Figure 3. Electrostatic persistence length P,as a function of chain length at different salt concentrations. bp is the total number of base pairs of the DNA. The DNA is modeled as a finite line of charges and the constant-dielectric model is employed for the solvent.
R values greater than 700 A (with the point (0,O)included since AG = 0 for (1/R)2= 0). Figure 2 shows P, as a function of the inverse of salt concentration. To a very good approximation P, varies linearly as a function of the inverse of the salt concentration over the whole salt region 10-3-1 M. (For the smallest salt concentration, M, the Debye length is 96.12 A; therefore, no electrostatic end effects occur since for the 600-mer the total contour length 2028 A is substantially larger than the Debye screening length.) Our prediction of a linear dependence of P, with the inverse of salt concentration is in agreement with the Debye-Htickel result for the infinite uniformly charged lime model obtained independently by Odijk16 and Skolnick and Fixman." We now examine how the electrostatic persistence length approaches the infinite chain limit. Figure 3 shows the chain length dependence of P, at different salt concentrations; P, increases with increasing chain length. The approach to a constant plateau value (16) Odijk, T. J. J . Polym. Sci., Polym. Phys. Ed. 1977, IS, 477-403. (17) Skolnick, J.; Fixman, M. Macromolecules 1977, 10, 944-948.
3966
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 400
Fenley et al. TABLE 11: Effect of Varying Radius of Curvature R on the Sodium Binding Fraction BI of a 600-Base-Pair DNA at Various Salt Concentrations c,"
e, 300
R = 2000 8,
R = 400 8,
c,, M
R = m (straight B-DNA)
(bent B-DNA)
(bent B-DNA)
0.0001 0.0010 0.0100 0.1000
0.6217 0.5713 0.4457 0.2 170
0.6221 0.5714 0.4457 0.2170
0.6294 0.5726 0.4460 0.2171
~~~
h
5 hW
200
"The helical axis of the straight DNA is smoothly bent into a circular arc of radius R. The DNA is modeled as a double helical distribution of charges. The constant dielectric (CD) model is adopted for the solvent.
100
0 O.ooO1 MNaCl
P
0.001 MNaCl 0
0
100
200
300
400
500
600
700
bP
Figure 4. Comparison between the chain length dependence of P, obtained with the finite line model with the Debye-Huckel results of Hagerman," at different salt concentrations. The upper curves correspond to 0.001 M NaCl and the lower curves to 0.01 M NaCI. The open symbols correspond to the line model results and the full (shaded) symbols to Hagerman's results.
of P, (polymer limit) is reached faster at higher salt concentrations. The polymer limit for the 0,0001 M curve is still not reached for the 600 bp DNA, even though the chain length 2028 8, of this fragment is substantially larger than the Debye screening length of 304 8,. This striking influence of electrostatic end effects on P, for DNA fragments that are much longer than the Debye screening length was pointed out in the Debye-Hiickel study of Hagerman." At all other salt concentrations considered, P, has essentially reached the infinite chain limit for a 600 bp chain length. Figure 4 compares our results with those of Hagerman; the two calculations, based on quite different procedures, but on similar models, are in semiquantitative agreement. According to these results, it is clearly important to take into consideration end effects when interpreting persistence length measurements of oligomeric DNA embedded in solutions of low salt concentrations. (b) Helical Model in a Constant Dielectric Solvent. We turn now to a more realistic treatment of the DNA molecule by considering the finite double-helical distribution of phosphate charges. We analyze first the constant dielectric model for the solvent; the dielectric saturation effect is addressed in the next section. To examine the role of electrostatic end effects on the persistence length, we consider DNA of varying lengths (60-600 base pairs). The effect of bending on the extent of Na+ binding is shown in Table 11. As discussed in an earlier paper (for straight DNA), the Na+ binding fraction is dependent on salt concentration, contrary to experimental measurements. As will be shown in the next section the salt dependence of the extent of counterion binding is suppressed when dielectric saturation effects are taken into account, and therefore, agreement with experimental data is reached. Again, as in the case of counterion binding fractions obtained with the finite line charge model, the extent of Na+ binding is virtually independent of the radius of curvature. Only a very slight increase of the Na+ charge fraction with decreasing radius of curvature is predicted at the low salt end (e, at R = 400 8, is reduced by about 1% relative to the Na+ binding fraction for straight DNA). It is important to examine when the Hookean behavior of the ionic bending free energy given by eq 7 breaks down for the double-helical charge model. Figure 5 addresses this question by showing the dependence on 1/R2 of the ionic bending free energy
0
2
4
6
8
Figure 5. Ionic bending free energy (AG)for a 600-base-pair DNA as a function of the square of the inverse of the radius of curvature ( R )at various salt concentrations. The DNA is modeled as a double-helical distribution of phosphate charges. The constant dielectric model is employed for the solvent.
of a 600-mer, at different salt concentrations. The strongest deviations from a linear correlation are observed for low salt; AC varies linearly with 1/R2 for R > 600 8, at all salt concentrations. The behavior is qualitatively similar to that of the line charge model (compare with Figure 1). We set out to examine the ionic strength dependence of the electrostatic persistence length for a 600 bp atomic level representation of DNA. Figure 6 shows P, as a function of the l o g arithm of salt concentration. It is clear that the variation is not linear either for monovalent or divalent counterions (the significance of this observation will be discussed). As expected, P, for the ion of higher charge (e.g., Mg2+)is smaller than that for Na+ throughout the entire salt range. This result is supported by experimental studies of Hagermanis and Maret and Weill.I9 The variation of P, as a function of the inverse of the square root of salt concentration is shown in Figure 7. To a good approximation the electrostatic persistence length is a linear function of cs4,sover the entire salt range considered (10-3-1 M). This trend is in agreement with the predictions made by Le Bret4 and Fixmad using the nonlinear PB equation for DNA modeled as a nonconducting cylinder; these authors also predict that for both monovalent and divalent counterions P, varies linearly with cs4,5 in the salt range 10-3-1 M . The inverse linear correlation of P, with the first power of c, for the line model (Figure 2 ) stands in (18) Hagerman, P. J. Biopolymers 1981, 20, 1503-1535. (19) Maret, G.; Weill, G . Biopolymers 1983, 22, 2727-2744.
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 3967
Electrostatic Persistence Length 8ooO
0 helicalmodel
h
‘d
h
4 m
5
e’
0 . ’
2000
C
\
-.
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0
-1
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1
log(cs )
-3
-2
1
0
1
200
Figure 6. Electrostatic persistence length P, for a 600-base-pair DNA as a function of the logarithm of salt concentration c,: (A)NaCl and (0) MgC12. The DNA is modeled as a double-helical distribution of phosphate charges. The constant dielectric model is employed for the
solvent. 1400
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Figure 7. Electrostatic persistence length P, as a function of the inverse of the square root of salt concentration c,. The dashed lines represent the least-squares fit of the results to a function of typey = ax + 6,where y and x represent P, and l/v‘c,, respectively: (0)NaCl and (A)MgCI2. The 600-mer DNA is modeled as a double-helical distribution of charges. The constant dielectric model is employed for the solvent.
contrast to this result, obtained for a more realistic distribution of polymer charges. Figure 8, top and bottom, compares our computed electrostatic persistence lengths for a 600 bp DNA immersed in NaCl and MgC12,respectively, with their counterparts obtained with the nonlinear PB equation for the infinite nonconducting ~ y l i n d e r . ~The . ~ two sets of results are qualitatively similar, although quantitative differences are also evident. Figure 9 shows the importance of end effects a t low salt; plots of P, for DNA fragments as a function of the number of base pairs in the fragment have the same qualitative appearance as Figure 3 for the line model and are not shown. In summary, there are two major differences between the persistence length results for the line model and the helical model, both in a constant dielectric medium. The electrostatic persistence lengths for the helical model are substantially larger than for the line model (this feature is most evident in Figure 11 below). Moreover, the salt dependences are qualitatively different; the
log (%)
Figure 8. Comparison between the electrostatic persistence length P, obtained by using the helical model with the PB results of Fixmad and Le Bret4in both NaCl (top) and MgC12(bottom) solutions. In our case,
we consider a W m e r DNA with the constant dielectric model employed for the solvent.
inverse linear correlation with salt for the line model is replaced by cn inverse correlation with the square root of salt for the helical model. (c) Incorporatioo of D i M c Saturation into the H elical Modd We know from previous resultsEthat inclusion of a distance-dependent dielectric constant generated by the saturation of dielectric polarization brings the properties of the helical model back into close conformance with those of the line model; in particular, the salt dependence of the binding fraction for the former is suppressed, and the characteristic salt-invariant binding fraction of the line model (and experiment) is restored. The same thing happens for the electrostatic persistence length. Because the salt invariance of the binding fraction currently seems like a strongly established experimental result, we believe that P, computed from the helical model with dielectric saturation represents our most reliable values. Table I11 shows that the binding fraction is essentially invariant to radius of curvature, a property shared with both the line model and the constant dielectric helical model. But the actual
Fenley et al.
3968 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 8000
300
250
.
I I
I
I
I
I
I
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b / / /
/ /
0 6oobp
m
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600
800
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-4
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F i i 10. Electrostatic persistence length P, as a function of the inverse of salt concentration c,. The dashed line corresponds to the least-squares fit of the results for NaCl to a function of typey = ax + b, where y and x correspond to P, and l/cs, respectively. The helical model and the
dielectric saturation model are employed for the 600-mer DNA and solvent, respectively. an,
Figure 9. Electrostatic persistence length P, as a function of the loga-
rithm of salt concentration c, at different chain lengths. The helical model and the constant dielectric model are employed for the DNA and solvent, respectively.
0 Borochov et a1 500
-
0 constant dielectric model
A dielectric saturation model
TABLE IU: Effect of Varying Radius of Curvature R on the Sodium Binding Fraction 8 , of a 600-Base-Pair DNA at Various Salt Concentrations c.D
4 c,,
M
0.0001 0.0010 0.0100 0.1000
R = m
R = 2000 A
(straight B-DNA)
(bent B-DNA)
0.8010 0.8058
0.8011 0.8059 0.7958
0.8032 0.8061
0.7360
0.7361
0.7958 0.7360
R = 400 A (bent B-DNA) 0.7959
“The helical axis of the straight DNA is smoothly bent into a circular arc of radius R. The DNA is modeled as a double-helical distribution of charges. The dielectric saturation (DS) model is employed for the solvent. values of 8, in Table 111 resemble those of Table I for the line model; the salt dependence seen in Table I1 for the constant dielectric helical model is suppressed when the helical model includes dielectric saturation. In Figure 10 we can see a linear correlation with inverse salt concentration for the dielectric saturation helical model. We recall Figure 2, which shows a similar trend for the line model, and Figure 7 for the constant dielectric helical model, which exhibits a different correlation. End effects in the dielectric saturation helical model are similar to what may be seen in Figure 3 for the line model and are not shown separately. Finally, in Figure 11 the similarity of values of P, for both line and dielectric-saturation helical models throughout the range of experimentally accessible salt concentrations is made evident. The experimental data in this figure, and their relation to our computations, are discussed in the next section. (a) Comparison with Experiment. Figure 11 shows our electrostatic persistence lengths computed for the various models analyzed. The values refer to 600-base-pair DNA, for which end effects are negligible over the salt range considered in this graph. Also shown are the values obtained from the light scattering measurements of Borochov et al.: corrected for excluded volume” (Po, the nonelectrostatic contribution to the persistence length, is taken as 300 A, since the measured values of total persistence (20) Manning, G. S. Biopolymers
1981, 20, 1751-1755.
1 -3
\
-2
-1
0
1
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Figure 11. Comparison between the theoretical electrostatic persistence length P, (obtained with different models) with the corrected experimental values of Borochov et aL2 A 600-base-pair DNA is considered in all different models.
length P,,where P, = Po + P, and AG, = AGO+ Ace,level off at about this constant value at high salt’). As has been discussed elsewhere: we regard these measured values as the best available; they have been largely corroborated by data from two other laboratories,21,22which we could also have used without changing our conclusions. The experimental values of P, are strongly correlated (correlation coefficient R = 0.989) to a straight line when plotted, as in Figure 11, against log c,. None of the computed curves have this functional form. Moreover, the values of P, from our most reliable model, the helical model of DNA with dielectric saturation, are much smaller than the measured values. We note that the PB results of Le Bret4 and Fixman5 also do not agree with the experimental data. None of the theoretical persistence lengths based on a smooth-bending model for DNA agree with the measured salt (21) Cairney, K.L.; Harrington, R. E. Biopolymers 1982, 21, 923-934. (22) Harpst, J. A.; Sobel, E.S . Biophys. J . 1983, 4 1 , 288a.
The Journal of Physical Chemistry, Vol. 96, NO. 10, 1992 3969
Electrostatic Persistence Length I I y = 1 4 0 9 1 -j261.91xR=P95026 I
900 I
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Figure 12. Total persistence length Pt obtained by Porschkelo as a function of the logarithm of salt concentration c,. The solid and dashed line represent best-fit straight lines through three data points that nearly lie on a line and for all four data points, respectively.
dependence. If we accept the data, the two possibilities are that D N A bends smoothly but the calculations are deficient, or the calculations are reasonably accurate but D N A does not bend smoothly. A theory based on a different model for D N A bending, which reproduces the observed logarithmic salt dependence, has been formulated? Recently, Porschkeiohas reported a set of persistence length data for short D N A fragments based on hydrodynamic measurements, which confirms the strong salt dependence in the range 10-3-10-'M found by Borochov et al.2 and others. In common with the Borochov (light-scattering)data, Porschke's data cannot be fit to current theoretical curves based on smooth bending models. His data can be inversely correlated with the square root of salt concentrationlobut cannot be brought into agreement with the Fixman-Le Bret curve; the two sets do not differ by a constant nonelectrostatic value Po. The situation becomes still worse when our own calculated values are considered; our most reliable set (the dielectric saturation results) is inversely correlated with the
first power of salt, not the square root. (Our constant dielectric values are correlated with the inverse square root of salt and are in better agreement with Porschke's data, but the constant dielectric model possesses a strong dependence of the bindmg fraction on salt, which is not found experimentally, and which we regard as a serious drawback in the model.) Figure 12 shows an aspect of Porschke's data, not reported by the author, that we believe to be of some importance. It portrays a plot of Porschke's persistence length values as a function of log c,. Although Porschke emphasizes a correlation with the inverse square root of c,, we see that it is also possible to fit his data logarithmically, in a manner quite similar to the light scattering data in Figure 11. Three of his four points fit a straight line with 99% correlation. Inclusion of the fourth point results in some deterioration of the linear fitting, but the correlation coefficient, 0.95, is still high; moreover, no systematic curvature away from the best-fit straight line is apparent. The line that fits the four points best has an equation, y = 141 - 262x (A units), similar to the equation, y = 317 - 216x, of the line drawn through the data points in Figure 11 (300A is added to each point to get the total persistence length); the slopes, in particular, differ by only 18%. The light scattering data (Figure 11) are correlated linearly with log c,, not with c , ~ , ~Porschke's . four data points are successfully correlated with both functional forms. A clearer characterization of his data must, in our opinion, await satisfaction of several criteria. A denser grid of points must be generated in the salt range considered. Electrostatic end effects for the D N A fragments used must be recognized, especially at the low salt end. If only a few fragments are used, those suspected of having sequences not typical of bulk D N A should be eliminated (for example, one of the fragments was reported by Porschke to have an atypical temperature dependence, possibly reflecting static curvature). In elaboration of this last point, we note that Porschke's persistence lengths represent an average over fragments of different lengths. Only a very few fragments are used. If the sample includes fragments of atypical sequence, the resulting value of the persistence length could be skewed away from that characteristic of bulk D N A . Of course, the persistence lengths of unusual sequences are also of interest but may not be directly comparable to the light scattering data.
